Emergent invariance and scaling properties in the collective return dynamics of a stock market.

A key metric to determine the performance of a stock in a market is its return over different investment horizons (τ). Several works have observed heavy-tailed behavior in the distributions of returns in different markets, which are observable indicators of underlying complex dynamics. Such prior works study return distributions that are marginalized across the individual stocks in the market, and do not track statistics about the joint distributions of returns conditioned on different stocks, which would be useful for optimizing inter-stock asset allocation strategies. As a step towards this goal, we study emergent phenomena in the distributions of returns as captured by their pairwise correlations. In particular, we consider the pairwise (between stocks i, j) partial correlations of returns with respect to the market mode, ci,j(τ), (thus, correcting for the baseline return behavior of the market), over different time horizons (τ), and discover two novel emergent phenomena: (i) the standardized distributions of the ci,j(τ)'s are observed to be invariant of τ ranging from from 1000min (2.5 days) to 30000min (2.5 months); (ii) the scaling of the standard deviation of ci,j(τ)'s with τ admits good fits to simple model classes such as a power-law τ-λ or stretched exponential function [Formula: see text] (λ, ß > 0). Moreover, the parameters governing these fits provide a summary view of market health: for instance, in years marked by unprecedented financial crises-for example 2008 and 2020-values of λ (scaling exponent) are substantially lower. Finally, we demonstrate that the observed emergent behavior cannot be adequately supported by existing generative frameworks such as single- and multi-factor models. We introduce a promising agent-based Vicsek model that closes this gap.

Quantum advantage in variational Bayes inference.

Variational Bayes (VB) inference algorithm is used widely to estimate both the parameters and the unobserved hidden variables in generative statistical models. The algorithm-inspired by variational methods used in computational physics-is iterative and can get easily stuck in local minima, even when classical techniques, such as deterministic annealing (DA), are used. We study a VB inference algorithm based on a nontraditional quantum annealing approach-referred to as quantum annealing variational Bayes (QAVB) inference-and show that there is indeed a quantum advantage to QAVB over its classical counterparts. In particular, we show that such better performance is rooted in key quantum mechanics concepts: i) The ground state of the Hamiltonian of a quantum system-defined from the given data-corresponds to an optimal solution for the minimization problem of the variational free energy at very low temperatures; ii) such a ground state can be achieved by a technique paralleling the quantum annealing process; and iii) starting from this ground state, the optimal solution to the VB problem can be achieved by increasing the heat bath temperature to unity, and thereby avoiding local minima introduced by spontaneous symmetry breaking observed in classical physics based VB algorithms. We also show that the update equations of QAVB can be potentially implemented using ⌈logK⌉ qubits and 𝒪(K) operations per step, where K is the number of values hidden categorical variables can take. Thus, QAVB can match the time complexity of existing VB algorithms, while delivering higher performance.

Ansatz-Independent Variational Quantum Classifiers and the Price of Ansatz.

The paradigm of variational quantum classifiers (VQCs) encodes classical information as quantum states, followed by quantum processing and then measurements to generate classical predictions. VQCs are promising candidates for efficient utilizations of noisy intermediate scale quantum (NISQ) devices: classifiers involving M-dimensional datasets can be implemented with only [Formula: see text] qubits by using an amplitude encoding. A general framework for designing and training VQCs, however, is lacking. An encouraging specific embodiment of VQCs, quantum circuit learning (QCL), utilizes an ansatz: a circuit with a predetermined circuit geometry and parametrized gates expressing a time-evolution unitary operator; training involves learning the gate parameters through a gradient-descent algorithm where the gradients themselves can be efficiently estimated by the quantum circuit. The representational power of QCL, however, depends strongly on the choice of the ansatz, as it limits the range of possible unitary operators that a VQC can search over. Equally importantly, the landscape of the optimization problem may have challenging properties such as barren plateaus and the associated gradient-descent algorithm may not find good local minima. Thus, it is critically important to estimate (i) the price of ansatz; that is, the gap between the performance of QCL and the performance of ansatz-independent VQCs, and (ii) the price of using quantum circuits as classical classifiers: that is, the performance gap between VQCs and equivalent classical classifiers. This paper develops a computational framework to address both these open problems. First, it shows that VQCs, including QCL, fit inside the well-known kernel method. Next it introduces a framework for efficiently designing ansatz-independent VQCs, which we call the unitary kernel method (UKM). The UKM framework enables one to estimate the first known computationally-determined bounds on both the price of ansatz and the price of any speedup advantages of VQCs: numerical results with datatsets of various dimensions, ranging from 4 to 256, show that the ansatz-induced gap can vary between 10 and 20[Formula: see text], while the VQC-induced gap (between VQC and kernel method) can vary between 10 and 16[Formula: see text]. To further understand the role of ansatz in VQCs, we also propose a method of decomposing a given unitary operator into a quantum circuit, which we call the variational circuit realization (VCR): given any parameterized circuit block (as for example, used in QCL), it finds optimal parameters and the number of layers of the circuit block required to approximate any target unitary operator with a given precision.

Many-body perturbation theory and fluctuation relations for interacting population dynamics.

Population dynamics deals with the collective phenomena of living organisms, and it has attracted much attention since it is expected to explain how not only living organisms but also human beings have been adapted to varying environments. However, it is quite difficult to insist on a general statement on living organisms since mathematical models heavily depend on phenomena that we focus on. Recently, it was reported that the fluctuation relations on the fitness of living organisms held for a quite general problem setting. But, interactions between organisms were not incorporated in the problem setting, though interaction plays critical roles in collective phenomena in physics and population dynamics. In this paper, we propose interacting models for population dynamics and provide the perturbative theory of population dynamics. Then, we derive the variational principle and fluctuation relations for interacting population dynamics.